Blocking light in compact Riemannian manifolds · Blocking light is also interesting in the context...

Geometry & Topology 11 (2007) 867–887 867

Blocking light in compact Riemannian manifolds

JEAN-FRANÇOIS LAFONT

BENJAMIN SCHMIDT

We study compact Riemannian manifolds .M; g/ for which the light from any givenpoint x 2 M can be shaded away from any other point y 2 M by finitely manypoint shades in M . Compact flat Riemannian manifolds are known to have this finiteblocking property. We conjecture that amongst compact Riemannian manifolds thisfinite blocking property characterizes the flat metrics. Using entropy considerations,we verify this conjecture amongst metrics with nonpositive sectional curvatures.Using the same approach, K Burns and E Gutkin have independently obtained thisresult. Additionally, we show that compact quotients of Euclidean buildings have thefinite blocking property.

On the positive curvature side, we conjecture that compact Riemannian manifolds withthe same blocking properties as compact rank one symmetric spaces are necessarilyisometric to a compact rank one symmetric space. We include some results providingevidence for this conjecture.

53C22; 53C20, 53B20

1 Introduction

To what extent does the collision of light determine the global geometry of space? In thispaper we study compact Riemannian manifolds with this question in mind. Throughout,we assume that .M; g/ is a smooth, connected, and compact manifold without boundaryequipped with a smooth Riemannian metric g . Unless stated otherwise, a geodesicsegment refers to a unit speed paramaterization of a geodesic in M . Notationally,we will write such a paramaterized segment as W Œ0; L � ! M; where L is thelength of the segment . By the interior of a geodesic segment we mean the setint. / WD ..0; L // � M .

Definition (Light) Let X; Y � .M; g/ be two nonempty subsets, and let Gg.X; Y /

denote the set of geodesic segments � M with initial point .0/ 2 X and terminalpoint .L / 2 Y . The light from X to Y is the set

Lg.X; Y / D f 2 Gg.X; Y / j int. / \ .X [ Y / D ∅g:

Published: 27 May 2007 DOI: 10.2140/gt.2007.11.867

http://www.ams.org/msnmain?fn=705&pg1=CODE&op1=OR&s1=53C22,(53C20, 53B20)

http://dx.doi.org/10.2140/gt.2007.11.867

868 Jean-François Lafont and Benjamin Schmidt

Definition (Blocking Set) Let X; Y � M be two nonempty subsets. A subset B isa blocking set for Lg.X; Y / provided that for every 2 Lg.X; Y /;

int. / \ B ¤ ∅:

In this paper we focus on compact Riemannian manifolds for which the light betweenpairs of points in M is blocked by a finite set of points. We remark that by a celebratedtheorem of Serre [24], Gg.x; y/ is infinite when x; y 2 M are two distinct points. Incontrast, Lg.x; y/ � Gg.x; y/ may or may not be an infinite subset. For example, inthe case of a round metric on a sphere, jL.x; y/j is infinite only when x D y or x andy are an antipodal pair.

Definition (Blocking Number) Let x; y 2 M be two (not necessarily distinct) pointsin .M; g/. The blocking number bg.x; y/ for Lg.x; y/ is defined by

bg.x; y/ D inffn 2 N [ f1g j Lg.x; y/ is blocked by n pointsg:

Definition (Finite Blocking Property) A compact Riemannian manifold .M; g/ issaid to have finite blocking if bg.x; y/ < 1 for every .x; y/ 2 M �M . When .M; g/

has finite blocking and the blocking numbers are uniformly bounded above, .M; g/ issaid to have uniform finite blocking.

The finite blocking property seems to have originated in the study of polygonal billiardsystems and translational surfaces (see Fomin [7], Gutkin [9; 10; 11], Heimer andSnurnikov [13], and Monteil [16; 17; 18; 19]). Our motivation comes from the followingtheorem (see [7], [9, Lemma 1], or Gutkin and Schroeder [12, Proposition 2]):

Theorem Compact flat Riemannian manifolds have uniform finite blocking.

We believe the following is true:

Conjecture Let .M; g/ be a compact Riemannian manifold with finite blocking.Then g is a flat metric.

There is a natural analogue of (uniform) finite blocking for general geodesic metricspaces. We provide an extension of the above theorem in Section 5:

Theorem 1 Compact quotients of Euclidean buildings have uniform finite blocking.

As evidence for the above conjecture we prove the following theorem in Section 4:

Geometry & Topology, Volume 11 (2007)

Blocking light in compact Riemannian manifolds 869

Theorem 2 Let .M; g/ be a compact nonpositively curved Riemannian manifold withthe finite blocking property. Then g is a flat metric.

This theorem is a consequence of a well known result about nonpositively curvedmanifolds and the next theorem relating the finite blocking property to the topologicalentropy of the geodesic flow:

Theorem 3 Let .M; g/ be a compact Riemannian manifold without conjugate points.If htop.g/ > 0, then bg.x; y/ D 1 for every .x; y/ 2 M . In other words, given any pairof points x; y 2 M and a finite set F � M � fx; yg, there exists a geodesic segmentconnecting x to y and avoiding F .

Working independently and using a similar approach K Burns and E Gutkin have alsoobtained Theorem 3 as well as the following [4]:

Theorem (Burns and Gutkin) Let .M; g/ be a compact Riemannian manifold withuniform finite blocking. Then htop.g/ D 0 and the fundamental group of M is virtuallynilpotent.

In Section 2, we define regular finite blocking by imposing a continuity and separationhypothesis on blocking sets. We show that manifolds with regular finite blocking haveuniform finite blocking and are conjugate point free. Combining this result with theprevious theorem of K Burns and E Gutkin, and recent work of N D Lebedeva [14]yields:

Theorem 4 Let .M; g/ be a compact Riemannian manifold with regular finite block-ing. Then g is a flat metric.

Blocking light is also interesting in the context of the nonnegatively curved compacttype locally symmetric spaces. In [12], Gutkin and Schroeder show the following:

Theorem (Gutkin and Schroeder) Let .M; g/ be a compact locally symmetric spaceof compact type with R-rank k �1. Then bg.x; y/�2k for almost all .x; y/2M �M .

We refer the reader to [12] for a more precise formulation and discussion of this result.On the positively curved side, we focus on the blocking properties of the compactrank one symmetric spaces or CROSSes. The CROSSes are classified and consist ofthe round spheres .Sn; can/, the projective spaces .KPn; can/ where K denotes oneof R,C, or H, and the Cayley projective plane .CaP2; can/ where “can” denotes asymmetric metric. The CROSSes all satisfy the following blocking property:



Definition (Cross Blocking) A compact Riemannian manifold .M; g/ is said to havecross blocking if

0 < d.x; y/ < Diam.M; g/ H) bg.x; y/ � 2:

Just as with finite blocking, we also define regular cross blocking by imposing acontinuity and separation hypothesis on blocking sets. In addition to cross blocking,round spheres also satisfy the following blocking property:

Definition (Sphere Blocking) A compact Riemannian manifold .M; g/ is said tohave sphere blocking if bg.x; x/ D 1 for every x 2 M .

This is a blocking interpretation of “antipodal points”; we think of the single blockerfor Lg.x; x/ as being antipodal to x . We believe the following is true:

Conjecture A compact simply connected Riemannian manifold .M; g/ has crossblocking if and only if .M; g/ is isometric to a simply connected compact rank onesymmetric space. In particular, .M; g/ has cross blocking and sphere blocking if andonly if .M; g/ is isometric to a round sphere.

As support for this conjecture, we prove the following theorems in Section 3:

Theorem 5 Let .S2; g/ be a metric on the two sphere with cross blocking and sphereblocking. Then a shortest periodic geodesic is simple with period 2 Diam.S2; g/.

Theorem 6 Let .M 2n; g/ be an even dimensional manifold with positive sectionalcurvatures and regular cross blocking. Then .M; g/ is a Blaschke manifold. If inaddition M is diffeomorphic to a sphere or has sphere blocking, then .M; g/ isisometric to a round sphere.

Theorem 7 Let .M; g/ be a compact Riemannian manifold with regular cross block-ing, sphere blocking, and which does not admit a nonvanishing line field. Then .M; g/

is isometric to an even dimensional round sphere.

Acknowledgements We owe our gratitude to Keith Burns; Keith pointed out a subtletyregarding Theorem 3 (Proposition 4.2 below) that we originally missed during theearlier stages of this work. We thank Ralf Spatzier for numerous helpful discussions.We would also like to thank Eugene Gutkin, Pilar Herreros and Tracy Payne for pointingout some inaccuracies in an early draft of this paper.



The first author would like to thank the Ecole Normale Superieure (Lyon) for theirhospitality. The work of the first author was partially supported by the National ScienceFoundation under grant DMS-0606002. The second author would like to thank TheOhio State University mathematics department for their hospitality. This research waspartially conducted during the period the second author was employed by the ClayMathematics Institute as a Liftoff Fellow.

2 Finite blocking and conjugate points

For the reader’s convenience, we begin with the definition of a conjugate point in acompact Riemannian manifold .M; g/. We let TM (resp. UM ) denote the tangentbundle (resp. unit tangent bundle) of M and denote the fibers above a point p 2 M ,by TpM and UpM . For a point p 2 M , the exponential map

exppW TpM ! M

is everywhere defined by completeness.

Definition (Conjugate Point) A point q D expp.v/ 2 M is conjugate to p along theunit speed geodesic vW Œ0; jjvjj� ! M with initial condition v

jjvjj2 Up.M / � UM if

d.expp/v is not of full rank.

In [25], F Warner describes the conjugate locus of singular points C.p/ � Tp.M /

for the exponential map expp . A point v 2 C.p/ is said to be regular if there existsa neighborhood U of v such that each ray emanating from the origin in Tp.M /

intersects at most one point in C.p/ \ U . The order of a point v 2 C.p/ is definedto be the dimension of the kernel of d.expp/v . Warner shows that the set of regularpoints C R.p/ � C.p/ is an open dense subset of C.p/ which (if nonempty) forms acodimension one submanifold of Tp.M /. Moreover, the order of points is constant ineach connected component of C R.p/ and there are normal forms (depending on theorder of the point) for the exponential map in a neighborhood of each regular point.From these normal forms, it follows that the preimage under the exponential map ofa regular conjugate point of order more than one is indiscrete. It appears that regularpoints of order more than one are rare in Riemannian manifolds (see Warner [26]).The next proposition shows that there are no such conjugate points in Riemannianmanifolds with the finite blocking property.

Proposition 2.1 Suppose that .M; g/ is a compact Riemannian manifold with finiteblocking. Then for each p 2 M , point preimages of expp are discrete subsets of TpM .



Proof Suppose not. Then there are points p; q 2 M and a sequence of vectors fvig �

exp�1p .q/ converging to a vector v1 2 exp�1

p .q/�Tp.M /. Let B Dfb1; : : : ; bkg�M

be a finite blocking set for Lg.p; q/. Note that any geodesic segment 2 Gg.p; q/

contains a subsegment 0 2 Lg.p; q/. It follows that for each index i , there is a welldefined time ti 2 .0; 1/, given by

ti D infft 2 .0; 1/ j expp.tvi/ 2 Bg:

After possibly relabeling blockers and passing to a subsequence, we may assume thatexpp.tivi/ D b1 for all i 2 N. A subsequence of the vectors ftivig converge to a vectort1v1 and by continuity of the exponential map, expp.t1v1/ D b1 . This shows thatthe point b1 is a sooner conjugate point to p along the geodesic ray .t/ D expp.tv1/

than is the point q . By repeating this argument, there is always a sooner conjugatepoint, contradicting the fact that conjugate points are discrete along a geodesic.

We expect that compact Riemannian manifolds with the finite blocking property willnever have conjugate points. Next we impose some restrictions on blocking setsand show that the light between conjugate points cannot be finitely blocked by suchsets. For a compact Riemannian manifold .M; g/, let M 0 � M � M be the subsetof points for which bg < 1, T 0 � TM be the subset of vectors .p; v/ 2 TM forwhich .p; expp.v// 2 M 0 , and let F.M / denote the set of finite subsets of M . Ablocking function for .M; g/ is a symmetric map BW M 0 ! F.M / such that for each.x; y/ 2 M 0; B.x; y/ is a finite blocking set for Lg.x; y/. Given a blocking functionB we define the first blocking time tBW T 0 ! .0; 1/ by

tB.p; v/ D infft 2 .0; 1/ j expp.tv/ 2 B.p; expp.v//g:

Definition (Continuous Blocking) We say that a closed Riemannian manifold .M; g/

has continuous blocking if there is a blocking function B for which the first blockingtime tBW T 0 ! .0; 1/ is continuous.

Definition (Separated Blocking) We say that a blocking function B is separated ifthere exists an � > 0 such that the � -neighborhoods of blocking points in each finiteblocking set B.x; y/ � M are disjoint.

Lemma 2.2 Let B be a separated blocking function for a compact Riemannian man-ifold .M; g/. Then the cardinalities of blocking sets defined by B are uniformlybounded above. In particular, compact Riemannian manifolds with finite blocking anda separated blocking function have uniform finite blocking.



Proof Suppose that the blocking sets defined by B are � -separated. An upper boundKmax for the sectional curvatures yields a lower bound C WD C.Kmax; �/ > 0 for thevolume of balls of radius � in M . Therefore, there are at most vol.M; g/=C disjointballs of radius � in M , concluding the proof.

Proposition 2.3 Let .M; g/ be a compact Riemannian manifold with a blockingfunction B that is both continuous and separated. Let p 2 M and suppose that U isan open subset of Tp.M / consisting of vectors from T 0 \ Tp.M /. Then no point inexpp.U / � M is a first conjugate point to the point p .

Proof Suppose not. Then there is a vector v 2 U for which expp.v/ is the firstconjugate point to p along the geodesic .t/ WD expp.tv/, t 2 Œ0; 1�. It is well known(see eg, Warner [25]) that expp is not one to one in any neighborhood of v . Let Bi

be a sequence of balls centered at v and contained in U with radii decreasing to zero.For each i , choose distinct points xi ; yi 2 Bi with expp.xi/ D expp.yi/ WD qi . LetB be a continuous and separated blocking function, and let li WD expp.tB.xi/xi/ andri WD expp.tB.yi/yi/ be the associated first blocking points in B.p; qi/. By continuityof B , the sequences flig and frig both converge to expp.tB.v/v/. By the separatednessof B , it follows that li D ri for all sufficiently large indices, whence expp.tB.v/v/ is asooner conjugate point along , a contradiction.

When a compact Riemannian manifold .M; g/ has finite (resp. cross) blocking anda continuous and separated blocking function, we shall say that .M; g/ has regularfinite (resp. regular cross) blocking.

Corollary 2.4 Let .M n; g/ be a compact Riemannian manifold with regular finiteblocking. Then .M; g/ has uniform finite blocking and is conjugate point free. Inparticular, the universal cover of M is diffeomorphic to Rn

Proof Since .M; g/ has finite blocking, T 0 D TM . By Lemma 2.2, .M; g/ hasuniform finite blocking and by Proposition 2.3, .M; g/ is conjugate point free. Thesecond statement is Hadamard’s theorem.

Corollary 2.5 Let .M; g/ be a compact Riemannian manifold with regular crossblocking. If p; q 2 M are consecutive conjugate points along a geodesic , thend.p; q/ D 0 or d.p; q/ D Diam.M; g/.

Proof Suppose p; q 2 M satisfy 0 < d.p; q/ < Diam.M; g/ and let W Œ0; 1� ! M

be a geodesic with .0/ D p , .1/ D q , and along which q is the first conjugate point



to p . Then .t/ D expp.tv/ for some v 2 Tp.M /: By continuity of the exponentialmap and the cross blocking property, there is an open set U � Tp.M / containingv and satisfying U � T 0 \ Tp.M /. By Proposition 2.3, the points p and q are notconsecutive conjugate points, a contradiction.

3 Blocking light and round spheres

In this section, we show under various hypothesis that compact Riemannian manifoldswith blocking properties similar to those of round spheres are necessarily isometric toround spheres. In general, we believe the following should be true:

Conjecture Let .M; g/ be a compact simply connected Riemannian manifold withcross blocking. Then .M; g/ is isometric to a simply connected compact rank onesymmetric space.

We begin by reviewing the definition and basic properties concerning cut pointsin a compact Riemannian manifold .M; g/. In this section, a unit speed geodesic W Œ0; L � ! M for which .0/ D .L / D p will be called a geodesic lasso based atp . For a geodesic lasso , we shall denote by �1W Œ0; L � ! M the geodesic lassoobtained by traversing in the reverse direction; specifically, �1.t/ WD .L � t/. Ifin addition the geodesic is regular at p , ie, P .0/ D P .L /, will be called a closedgeodesic based at p . By a simple lasso (resp. simple closed geodesic) based at p wemean a lasso (resp. closed geodesic) W Œ0; L � ! M based at p which is injective onthe interval .0; L / and with p … ..0; L //.

Definition Let .M; g/ be a compact Riemannian manifold, p 2 M , v 2 Up.M /, and W Œ0; 1/ ! M the unit speed geodesic ray defined by .t/ D expp.tv/ . Let Œ0; t0�

be the largest interval for which t 2 Œ0; t0� implies d.p; .t// D t . The point .t0/ issaid to be a cut point to p along the geodesic . The union of the cut points to p

along all the geodesics starting from p is called the cut locus and will be denoted byCut.p/.

The next two propositions are well known and describe points in the cut locus (see eg,do Carmo [6]).

Proposition 3.1 Suppose that .t0/ is the cut point of p D .0/ along a geodesic .Then either:

� .t0/ is the first conjugate point of .0/ along , or



� there exists a geodesic � ¤ from p to .t0/ such that length.�/ D length. /.

Conversely, if either of these two conditions is satisfied, then there exists t 0 2 .0; t0�

such that .t 0/ is the cut point of p along .

Proposition 3.2 Let p 2 M and suppose that q 2 Cut.p/ satisfies d.p; q/ D

d.p; Cut.p//. Then either:

� there exists a minimizing geodesic from p to q along which q is conjugate top , or

� there exist exactly two minimizing geodesics and � from p to q that togetherform a simple geodesic lasso based at p of length 2d.p; Cut.p//.

It follows from Proposition 3.1 that expp is injective on a ball of radius r centered atthe origin in Tp.M / if and only if r < d.p; Cut.p//.

Definition The injectivity radius of .M; g/ is defined to be

inj.M; g/ D infp2M

d.p; Cut.p//:

Note that the injectivity radius of a compact Riemannian manifold is never larger thanits diameter. Compact manifolds for which the injectivity radius equals the diameterare known as Blaschke manifolds. All of the compact rank one symmetric spacesare Blaschke and the well known Blaschke conjecture asserts that these are the onlyBlaschke manifolds of dimension at least two. We will use the following theorem fromBerger [2, Appendix D], extending earlier work of Green [8]:

Theorem 3.3 (Berger) Let .M; g/ be a Blashke metric on a smooth sphere. Thenthe metric g is a symmetric metric.

For a Blaschke manifold .M; g/, all infinite geodesics W R ! M cover simple closedgeodesics of length 2 Diam.M; g/ (see [2, Corollary 5.42]). The next proposition is afirst step in showing that geodesics in a manifold with cross blocking behave similarlyto those in a Blaschke manifold.

Proposition 3.4 Suppose that .M; g/ has cross blocking. Let W Œ0; L � ! M be aunit speed simple geodesic lasso based at p 2 M . Then L � 2 Diam.M; g/ and thepoint .L =2/ is the cut point to p in both of the directions P .0/ and � P .L /.



Proof Let D WD Diam.M; g/ and let c1 be the cut point to p in the direction P .0/

and c2 be the cut point to p in the direction � P .L /. By simplicity of , thereexists unique t1; t2 2 .0; L / such that ci D .ti/ for i D 1; 2. By Proposition 3.1,t1 � L =2 � t2 . The statement of the proposition follows from showing that t1 D t2 ,ie, that c1 D c2 and the point .L =2/ is the cut point to p in both of the directionsP .0/ and � P .L /.

We now will assume that t1 < t2 and must obtain a contradiction. With this assumption,we first argue that d.p; ..t1; t2/// D D . If not, choose a t 2 .t1; t2/ for which0 < d.p; .t// < D . Note that since is simple, the restrictions of to the intervalŒ0; t � and �1 to the interval Œ0; L � t � define distinct elements in Lg.p; .t// withnonintersecting interiors. There must be a single blocking point on the restrictionof to the interior of each of these intervals since .M; g/ has cross blocking. Asneither of these light rays are minimizing, there is a unit speed minimizing geodesic� W Œ0; L� � ! M joining p to .t/. Since � is minimizing, it defines a third lightray between p and .t/, whence its interior must intersect one of the two blockingpoints for Lg.p; .t//. Let s0 WD inffs 2 .0; L� / j �.s/ 2 int. /g. By simplicity of ,there is a unique t 0 2 .0; L / such that �.s0/ D .t 0/ WD q . Since � is unit speed andminimizing, 0 < d.p; q/ D s0 < L� D d.p; .t// < D , so that bg.p; q/ � 2 by the crossblocking condition. However, the restriction of to Œ0; t 0�, �1 to Œ0; L � t 0�, and therestriction of � to Œ0; s0� define three distinct elements in Lg.p; q/ with nonintersectinginteriors, implying bg.p; q/ � 3. This is a contradiction, whence d.p; ..t1; t2/// D D .

Next we show that d.p; ..t1; t2/// D D yields a contradiction, completing the proof.By the discreteness of conjugate points along geodesics, there is a t 2 .t1; t2/ so thatq WD .t/ is not conjugate to p in the direction P .0/ or in the direction � P .L /.Neither of the restrictions of to Œ0; t � or �1 to Œ0; L � t � is minimizing so that thereis a unit speed minimizing geodesic � W Œ0; D� ! M joining p to q . Note that since �

is minimizing, the interior of � cannot intersect . Indeed, a first point of intersectionbetween the interiors of � and would be at a point p0 satisfying 0 < d.p; p0/ < D sothat the reasoning from the previous paragraph may be applied to obtain a contradiction.Let qn D�.D�1=2n/. Choose sufficiently small neighborhoods B1 of t P .0/ and B2 of�.L � t/ P .L / on which expp restricts to a local diffeomorphism. For all sufficientlylarge n, there are unique xn 2 B1 and yn 2 B2 such that qn D expp.xn/ D expp.yn/.It follows by the continuity properties of the exponential map that for suitably large n,the geodesics

s 7! expp.sxn/ and s 7! expp.syn/

for s 2 Œ0; 1� and the restriction of � to the interval Œ0; D �1=2n� define three light raysbetween p and qn with nonintersecting interior. Hence, bg.p; qn/�3 for suitably largen. But 0 < d.p; qn/ < D , so that bg.p; qn/ � 2 by cross blocking, a contradiction.



Lemma 3.5 Suppose that .M; g/ has cross blocking and sphere blocking. Supposethat W Œ0; L � ! M is a unit speed simple geodesic lasso based at p 2 M . IfL < 2 Diam.M; g/, then is regular at p and all lassos based at p finitely cover . If L D 2 Diam.M; g/, then the interior of all of the geodesic lassos through p

intersect in the point .L =2/.

Proof Let p WD .L =2/. Suppose there is a (not necessarily simple) unit speed lasso� W Œ0; L� � ! M through p with P�.0/ distinct from P .0/ and � P .L /. As .M; g/ hassphere blocking and is simple, there is a unique t 2 .0; L / such that .t/ blocksLg.p; p/. Let s WD infft 2 .0; L� � j �.t/ D pg. The restriction of � to the interval Œ0; s�

gives an element in Lg.p; p/ so that by sphere blocking, its interior must pass throughthe blocking point .t/ (and hence int. /). Let s0 WD infft 2 .0; s/ j �.t/ 2 int. /g. Bysimplicity of there is a unique t 0 2 .0; L / such that .t 0/ D �.s0/ WD q . As issimple, the restrictions of to the intervals Œ0; t 0�, �1 to the interval Œt 0; L �, and � tothe interval Œ0; s0� define three distinct light rays between p and q with nonintersectinginteriors. Since p ¤ q cross blocking implies that d.p; q/ D Diam.M; g/ and that q D

p (by Proposition 3.4 and since L =2 � Diam.M; g/). Hence, if L =2 < Diam.M; g/

there are no geodesic lassos through p with initial tangent vector outside of the setf P .0/; � P .L /g, and if L =2 D Diam.M; g/, the interior of any lasso through thepoint p passes through the point p . This concludes the proof of the last statement inthe lemma.

We now assume that L =2 < Diam.M; g/, and will argue that is regular at the pointp and that all lassos at p finitely cover . By simplicity of , the restriction of

to the intervals Œ0; L =2� and �1 to the interval Œ0; L =2� define distinct light raysbetween p and p with nonintersecting interiors. As .M; g/ has cross blocking, theinterior of a third light ray must intersect the interior of in a blocker and will thereforehave a first point of intersection p0 with the interior of . This implies bg.p; p0/ � 3,a contradiciton. Therefore jLg.p; p/j D 2, while Gg.p; p/ is infinite by [24]. Notethat any geodesic segment from Gg.p; p/ � Lg.p; p/ is obtained from extending oneof the two light rays in Lg.p; p/. Each such extension gives rise to a geodesic lassobased at p with initial tangent vector in the set f� P .0/; P .L /g. But by the previousparagraph, the initial tangent vector of all lassos at p lie in the set f P .0/; � P .L /g:

Therefore, must be regular at p and all lassos at p finitely cover .

Definition A SC2L manifold is a Riemannian manifold with the property that allgeodesics cover simple closed geodesics of length 2L.

It is tempting to think that the only SC2L manifolds are the CROSSes. Amazingly,O Zoll exhibited an exotic SC2L real analytic Riemannian metric on the two sphere



[27]. This example is discussed in [2, Chapter 4] along with examples on higherdimensional spheres. We remark that these examples are not cross blocked. Indeed,Proposition 3.4 implies that SC2L manifolds with cross blocking are Blaschke withinj.M; g/ D Diam.M; g/ D L, while Theorem 3.3 asserts that there are no exoticBlaschke metrics on spheres. In view of our conjecture that the manifolds with crossblocking are precisely the CROSSes and the Blaschke conjecture that the Blaschkemanifolds are precisely the CROSSes, we expect that Blaschke manifolds are preciselythose manifolds with cross blocking. In the next proposition, we use well knownresults concerning Blaschke manifolds to show that Blaschke manifolds all have crossblocking.

Proposition 3.6 Suppose that .M; g/ is a Blaschke manifold. Then .M; g/ has crossblocking.

Proof Suppose that p; q 2 M satisfy 0 < d.p; q/ < D WD Diam.M; g/. By [2,Corollary 5.42], .M; g/ is a SC2D manifold. It follows that there is a unit speedsimple closed geodesic W Œ0; 2D� ! M with .0/ D p and .d.p; q// D q . Therestriction of to the intervals Œ0; d.p; q/� and �1 to Œ0; 2D � d.p; q/� give twodistinct elements in Lg.p; q/ with nonintersecting interiors. Hence bg.p; q/ � 2. Ifbg.p; q/ > 2 then there must be a third unit speed light ray ˇW Œ0; Lˇ � ! M joiningp to q . Note that since ˇ is a light ray and since all geodesics are periodic withperiod 2D , Lˇ < 2D . Moreover, since d.p; q/ < D D inj.M; g/, the restriction of to the interval Œ0; d.p; q/� is the unique minimizing geodesic from p to q so thatLˇ > inj.M; g/ D D . Extend ˇ to the simple closed geodesic ˇW Œ0; 2D� ! M . Thenthe restriction of ˇ to the interval ŒLˇ; 2D� gives a geodesic joining q to p of length2D � Lˇ < D D inj.M; g/. Hence, there are two minimizing geodesics joining p

and q , a contradiction. Therefore, bg.p; q/ D 2 and Blaschke manifolds have crossblocking.

Corollary 3.7 Suppose that .M; g/ is Blaschke manifold with sphere blocking. Then.M; g/ is isometric to a round sphere.

Proof By the last proposition .M; g/ has cross blocking. By Lemma 3.5, the interiorof all of the simple closed geodesics through p intersect in a single point p0 satisfyingd.p; p0/ D Diam.M; g/. By Proposition 3.4, p0 is the cut point to p along all thesegeodesics. Hence, dim.Cut.p// D 0, from which it follows (see eg [2, Proposition5.57]) that M is diffeomorphic to a sphere. By Theorem 3.3, .M; g/ is a roundsphere.

For the proof of the next theorem, we will need the following two definitions:



Definition For p 2 M and p0 2 Cut.p/, define the link ƒ.p; p0/ � Up0M by

ƒ.p; p0/Df� P .d.p; p0// j is a unit speed and minimizing geodesic from p to p0g:

Definition For p 2 M and U � UpM , U is said to be a great sphere if U is theintersection of a linear subspace of TpM with UpM .

Theorem 3.8 Suppose that .M; g/ is a compact Riemannian manifold with regularcross blocking, sphere blocking, and which does not admit a nonvanishing line field.Then .M; g/ is isometric to an even dimensional round sphere.

Proof First we argue that .M; g/ is a Blaschke manifold.

To obtain a contradiction, suppose that inj.M; g/ < Diam.M; g/ WD D . We begin byshowing that for each point x 2 M satisfying d.x; Cut.x// < D there is a uniquesimple closed geodesic based at x and this geodesic has length 2d.x; Cut.x//. Indeed,let x satisfy d.x; Cut.x// < D and x0 2 Cut.x/ satisfy d.x; x0/ D d.x; Cut.x//. ByCorollary 2.5 the points x and x0 are not conjugate so that by Proposition 3.2 there isa simple geodesic lasso C through x of length 2 inj.M; g/. By Lemma 3.5, C is asimple closed geodesic through x and is the unique lasso through x , as required.

Let L WD supfd.p; Cut.p// j p 2 M g � D . If L < D , the preceeding paragraph showsthat there is a unique closed geodesic Cp of length less than 2D through each pointp 2M: Since all of the Cq have lengths uniformly bounded above, whenever a sequenceof points fpig converge to a point p1 2 M , the sequence of closed geodesics Cpi

converge to a closed geodesic C1 . By the uniqueness of these geodesics, C1 D Cp1.

Therefore, the tangent spaces to these geodesics define a nonvanishing line field onM , a contradiction.

Hence, there is a point p 2 M satisfying d.p; Cut.p// D D . Such a point is saidto have spherical cut locus at p [2, Definition 5.22]. By [2, Proposition 5.44], thelink ƒ.p; p0/ � Up0.M / is a great sphere for each p0 2 Cut.p/, whence all geodesicsthrough p are periodic of period 2D . Now consider a geodesic connecting p to a pointq satisfying d.q; Cut.q// < D . This geodesic gives rise to a closed geodesic of length2D through q , while the first paragraph shows that there is a closed geodesic throughp of length 2d.q; Cut.q//. This contradicts Lemma 3.5, implying that at every pointq 2 M , we have d.q; Cut.q// D D , and hence concluding the proof that .M; g/ isBlaschke.

By Corollary 3.7, .M; g/ is isometric to a round sphere. As M does not admit anonvanishing line field, .M; g/ is isometric to an even dimensional round sphere.



Next, we adapt Klingenberg’s estimate on the injectivity radius to obtain the following(see [6, Chapter 13, Proposition 3.4]):

Theorem 3.9 Suppose that .M 2n; g/ is an even dimensional, orientable, Riemannianmanifold with positive sectional curvatures. If .M; g/ has regular cross blocking, then.M; g/ is Blaschke. In particular, if M is diffeomorphic to a sphere or .M; g/ hassphere blocking, then g is a round metric on a sphere.

Proof Suppose to the contrary that inj.M; g/ < Diam.M; g/ and choose p; q 2 M

so that q 2 Cut.p/ and d.p; q/ D inj.M; g/. By Corollary 2.5, p and q are notfirst conjugate points along any geodesic. By Propositions 3.1 and 3.2, there is aunit speed simple closed geodesic C W Œ0; 2 inj.M; g/� ! M of length 2 inj.M; g/

passing through p D C.0/ and q D C.inj.M; g//. Since M is orientable and evendimensional, parallel transport along C leaves invariant a vector v orthogonal to C

at C.0/. The field v.t/ along C.t/ is the variational field of closed curves Cs.t/

for s 2 Œ0; �/. As the sectional curvatures are strictly positive, the second variationalformula implies that length.Cs/ < length.C / for all small s > 0. For each s > 0, letqs be a point of Cs at maximum distance from Cs.0/. Necessarily, lims!0 qs D q andd.qs; Cs.0// < inj.M; g/. For each s > 0, let s be the unique minimizing geodesicjoining qs to Cs.0/. Note that each P s.0/ is orthogonal to Cs by the first variationalformula. Let w 2 TqM be an accumulation point of the vectors P s.0/ 2 Tqs

M

and W Œ0; 1� ! M be the geodesic defined by .t/ WD expq.tw/. It follows that

is a minimizing geodesic joining q to p which is orthogonal to C at q . As isminimizing, it cannot intersect C except at the points p and q , whence bg.p; q/ � 3, acontradiction. Therefore .M; g/ is Blaschke. The last statement follows from Theorem3.3 and Corollary 3.7.

Theorem 3.10 Suppose that .S2; g/ is a Riemannian metric on the two sphere withcross blocking and sphere blocking. Then a shortest nontrivial closed geodesic is simpleand has length 2 Diam.S2; g/.

Proof Let D WD Diam.M; g/ and let C be a shortest nontrivial closed geodesic. Wefirst argue that if C is simple, then its length is 2 Diam.S2; g/. Indeed, by Proposition3.4, length.C / � 2D . We suppose that length.C / < 2D and will obtain a contradiciton.Note that C separates S2 into two components. By Santalo’s formula (see [23, page488] or [1, page 290]), almost all geodesic rays with initial point on C eventuallyleave the component they initially enter. Choose one such ray W Œ0; 1/ ! M and lett WD infft 2 .0; 1/ j .t/ 2 C g: If .t/ is distinct from .0/, then bg. .0/; .t// � 3,a contradiction. Hence, .0/ D .t/, also a contradiction by Lemma 3.5.



Next we argue that a shortest nontrivial closed geodesic C is simple. By Nabutovskyand Rotman [20] or Sabourau [22], length.C / � 4D . Suppose that C is not simple,and choose a unit speed paramaterization C W Œ0; LC � ! M such that C.0/ is a crossingpoint. Let s D infft 2 .0; LC / j C.s/ D C.0/g. Without loss of generality, we mayassume that the restriction of C to the interval Œ0; s� defines a simple lasso at C.0/,whence s D 2D by Proposition 3.4. The restriction of C to the interval Œ2D; LC �

defines another lasso at C.0/. If this lasso is not simple, then it contains a simple lassoof length 2D and LC > 4D , a contradiction. Hence, the restriction of C to the intervalŒ2D; LC � defines a simple lasso and LC D 4D . By Lemma 3.5, C.D/ D C.3D/. Notethat the restriction of C to Œ0; 2D� separates S2 into two components. This implies thatC..0; 2D// \ C..3D; 4D// ¤ ∅. Letting s D infft 2 .3D; 4D/ j C.t/ 2 C..0; 2D//g;

it follows that bg.C.D/; C.s// � 3, a contradiction.

4 Finite blocking property and entropy

In this section we relate the finite blocking property for a compact Riemannian manifold.M; g/ to the topological entropy htop.g/ of its geodesic flow.

Our starting point is a well known theorem (see eg, Mane [15, Corollary 1.2]) identifyingthe topological entropy with the exponential growth rate of the number of geodesicsbetween pairs of points in M . For x; y 2 M and T > 0, let nT .x; y/ (resp. mT .x; y/)denote the number of geodesic segments (resp. light rays) between the points x and y

of length no more than T .

Theorem 4.1 (Mañé) Let .M; g/ be a compact Riemannian manifold without conju-gate points. Then

htop.g/ D limT !1

log.nT .x; y//

T;

for all .x; y/ 2 M � M:

The main observation of this section lies in the following:

Proposition 4.2 Let .M; g/ be a compact Riemannian manifold without conjugatepoints. If htop.g/ > 0, then bg.x; y/ D 1 for all .x; y/ 2 M � M:

Proof Let I WD inj.M; g/. We first argue that nT .x; y/ � .T =2I/2 mT .x; y/. For aunit speed geodesic W Œ0; L � ! M in Gg.x; y/, let t1. / WD supft 2 Œ0; L / j .t/ D

xg and t2. / WD infft 2 .t1. /; L � j .t/ D yg. Restrict to the interval Œt1. /; t2. /�

and change the parameter of this interval to define a unit speed geodesic

Light. /W Œ0; t2. / � t1. /� ! M



in Lg.x; y/. Since for each ˇ 2 Lg.x; y/, Light.ˇ/ D ˇ , it follows that

LightW Gg.x; y/ ! Lg.x; y/

defines a surjective map. To conclude this step, it suffices to show that given a fixedunit speed ˇ 2 Lg.x; y/ of length not more than T > 0, there are at most .T =2I/2

distinct preimages of ˇ under the map Light of length not more than T . For each unitspeed geodesic W Œ0; L � ! M of length not more than T satisfying Light. / D ˇ ,P .t1. // D P.0/ and P .t2. // D P.Lˇ/ since geodesics are determined by their initialconditions. It follows that the number of preimages of ˇ having length bounded aboveby T coincides with the number of different extensions of ˇ to a unit speed geodesicˇ 2 Gg.x; y/ having length not more that T . Given such an extension ˇ , let n

ˇ.x/

and nˇ.y/ be the number of returns to x and the number or returns to y . Necessarily,

nˇ.x/; n

ˇ.y/ � T =2I since each return to x or return to y increases the length of ˇ

by at least 2I . Hence, by uniqueness of geodesics, nT .x; y/ � .T =2I/2 mT .x; y/.

To complete the proof, we argue by contradiction, assuming there is a pair of pointsx; y 2 M with a finite blocking set F D fb1; : : : bkg � M � fx; yg for Lg.x; y/. Bydefinition, the interior of any light ray 2 Lg.x; y/ passes through some point bi 2 F ,breaking into two geodesic segments 1 2 Gg.x; bi/ and 2 2 Gg.bi ; y/. If haslength bounded above by T , then one of 1 or 2 must have length bounded above byT =2. Moreover, given a geodesic segment ˛ 2 Gg.x; bi/ (resp. ˇ 2 Gg.y; bi/), thereis at most one extension of ˛ (resp. ˇ ) to a light ray between x and y . It follows thatmT .x; y/ �

PkjD1 nT =2.x; bj / C nT =2.bj ; y/. Combining this with the estimate from

the previous paragraph yields:

nT .x; y/ � .T =2I/2kX

jD1

nT =2.x; bj / C nT =2.bj ; y/:

Let 0 < � < htop.g/=3. By Theorem 4.1, there is a T0 2 R so that T > To implies

jhtop.g/ �log.nT .�1; �2//

Tj < �;

for all �1; �2 2 fx; yg [ F: Therefore

exp.htop.g/��/T < nT .�1; �2/ < exp.htop.g/C�/T ;

for all T > T0 and �1; �2 2 fx; yg [ F . It now follows that

exp.htop.g/��/T < nT .x; y/ < 2k.T =2I/2 exp.htop.g/C�/T =2;

a contradiction for all sufficiently large values of T .



We remark that the conclusion bg.x; y/ D 1 for all .x; y/ 2 M �M may be phrasedmore geometrically as saying that given any point .x; y/ 2 M � M and any finite setF � M � fx; yg, there is a geodesic segment between x and y avoiding F .

As a corollary of Proposition 4.2, we obtain the following:

Theorem 4.3 Let .M; g/ be a compact Riemannian manifold with nonpositive sec-tional curvatures. Then .M; g/ has finite blocking if and only if .M; g/ is flat.

Proof Assume that .M; g/ has nonpositive curvature and is not flat. Then .M; g/

has no conjugate or focal points. By Pesin [21, Corollary 3], a geodesic flow on anonflat compact Riemannian manifold without focal points has positive entropy. ByProposition 4.2, .M; g/ does not have finite blocking.

K Burns and E Gutkin [4] made the nice observation that by assuming uniform finiteblocking and by iterating the line of reasoning used in the proof of Proposition 4.2 onecan establish the following:

Theorem 4.4 (Burns-Gutkin) Let .M; g/ be a compact Riemannian manifold withthe uniform finite blocking. Then htop.g/ D 0 and �1.M / has polynomial growth.

Using their result we obtain the following:

Theorem 4.5 Let .M; g/ be a compact Riemannian manifold with regular finiteblocking. Then .M; g/ is flat.

Proof By Corollary 2.4, .M; g/ has uniform finite blocking and is conjugate pointfree. By Theorem 4.4, �1.M / has polynomial growth. By Lebedeva [14], compactRiemannian manifolds without conjugate points and with polynomial growth funda-mental group are flat.

5 Finite blocking property and buildings

In this section, we provide a proof of Theorem 1, which states that compact quotientsof Euclidean buildings have uniform finite blocking. Let us start by recalling someelementary facts about Euclidean buildings, referring the reader to Brown [3] for moredetails.

Let W � Rn be a compact polyhedron, with all faces forming angles of the form �=mij

for some positive integer mij . Let ƒ � Isom.Rn/ be the Coxeter group generated by



reflections in the faces of the polyhedron, and observe that the ƒ-orbit of W generatesa tessellation of Rn by isometric copies of W . We can label the faces of the copies ofW in the tessellation of Rn according to the face of W whose orbit contains them.A Euclidean building is a polyhedral complex zX , equipped with a CAT(0)-metric,having the property that each top dimensional polyhedron is isometric to W (thesewill be called chambers). In addition, a certain number of axioms are required to besatisfied. We omit a precise definition of Euclidean buildings, contenting ourselveswith mentioning the properties we will need. The reader may refer to [3] for a precisedefinition, and to Davis [5] for geometric properties of these buildings. The polyhedralcomplex must also satisfy:

� Each face of the complex zX is labelled with one of the faces of the polyhedronW .

� Given any pair of points x; y 2 zX , there exists an isometric, polyhedral, labelpreserving embedding of the tessellated Rn whose image contains x and y . Theimage of such an embedding is called an apartment.

� The group Isom. zX / is defined to be the group of label preserving isometries ofzX .

� Given any two apartments A1;A2 whose intersection is non-empty, there existsan element � 2 Isom. zX / which fixes pointwise A1 \A2 , and satisfies �.A1/ D

A2 .

We will say that X is a compact quotient of zX provided it is the quotient of zX by acocompact subgroup of Isom. zX /, acting fixed point freely.

Note that in a Euclidean building, one has uniqueness of geodesics joining pairsof points (from the CAT(0) hypothesis). Furthermore, we can pick an apartmentcontaining both x and y , giving a totally geodesic Rn inside zX containing x; y . Thenthe geodesic joining x to y coincides with the straight line segment from x to y

within the apartment. We will call the point along the geodesic that is equidistant fromx and y the midpoint of x and y , and denote it by .x C y/=2.

Another important point is that both the building zX , as well as the compact quotientX come equipped with a canonical folding map to the canonical chamber W , givenby the labeling. We will use � to denote the canonical folding map, and given a pointx 2 zX (or in zX= � ), we define the type of the point x to be the point �.x/ 2 W . Wenow make two observations:

� In a compact quotient zX= � , there are only finitely many points of any giventype.



� Given any point p in zX = � , every pre-image of p in the universal cover zX hasexactly the same type as p .

The proof of the theorem will make use of the following easy lemma:

Lemma 5.1 Let A be any apartment, and x; y 2 W a pair of points in the modelchamber. Define S.x/, S.y/ to be the set of points in A of type x , y respectively.Then there exist a finite collection of points b1; : : : ; bk 2 W having the property that:

f.xx C xy/=2 j xx 2 S.x/; xy 2 S.y/g �

k[iD1

S.bi/

Proof We first observe that the Coxeter group ƒ contains an isomorphic copy of Zr

as a finite index subgroup, where r D dim.A/. In particular, if we denote by ƒ0 thisfinite index subgroup, we note that each of the two sets S.x/, S.y/ are the union ofŒƒ W ƒ0� D m disjoint copies of ƒ0 -orbits in A . Now note that given any two ƒ0 -orbitsin A, the collection of midpoints of points in the first orbit with points in the secondorbit lie in a finite collection of ƒ0 -orbits (in fact, at most 2r such orbits). This isimmediate from the proof of the fact that a flat torus has finite blocking.

This in turn implies that the collection of midpoints of points in the set S.x/ and pointsin the set S.y/ lie in the union of at most 2r m2 of the ƒ0 -orbits in A. Since eachƒ0 -orbit lies in a corresponding ƒ-orbit, we conclude that the collection of midpointslie in the union of a finite collection of ƒ-orbits. But two points are in the same ƒ-orbitif and only if they have the same type. Hence choosing the points b1; : : : ; bk to be thefinitely many types (at most 2r m2 of them), we get the desired containment of sets.

We now proceed to prove Theorem 1:

Proof of Theorem 1 Let X D zX= � be a compact quotient of the Euclidean buildingzX , and let W denote a model chamber. Given two points x; y in the space X , we want

to exhibit a finite set of blockers. Consider the sets P.x/;P.y/ � zX consisting of allpre-images of the points x; y , respectively, under the covering map zX ! X D zX= � .As we previously remarked, we can make sense of the midpoint of a pair of points inzX . We now claim that the collection of midpoints joining points in P.x/ to points inP.y/ have only a finite number of possible types in W .

In order to see this, let us apply the previous lemma to the types �.x/; �.y/ 2 W .First note that given an arbitrary pair of points xx 2 P.x/, xy 2 P.x/, there existsan apartment A containing xx; xy . Furthermore, we have the obvious containments



P.x/ \A� S.�.x//, P.y/ \ A � S.�.y//, and hence the midpoint joining xx to xy

has the property that its type is one of the finitely many points b1; : : : ; bk .

Hence to obtain a finite blocking set, let us consider the collection of all points inX whose type is one of b1; : : : ; bk . This yields a finite collection B of points in X

having the property that if is an arbitrary geodesic joining x to y , its midpoint mustbe one of the points in B , completing the proof of the theorem.

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Department of Mathematics, The Ohio State UniversityColumbus, OH 43210, USA

Department of Mathematics, University of Chicago5734 S University Avenue, Chicago, IL 60637, USA

[email protected], [email protected]

Proposed: Benson Farb Received: 3 August 2006Seconded: Walter Neumann, Tobias Colding Accepted: 21 March 2007


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mailto:[email protected]

mailto:[email protected]

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